I believe that the main component that adds stability to the max payload package is the LT tires. If you order a 4wd drive F150 you can specify the LT tires as an option for $300 on any version. For the max tow rating you have to have 20" wheels and those are not available with max payload. It is probable that the larger rims add stability too and you can retrofit LT tires to them, it just costs more. The only other features that would add stability are the springs and shocks and you can always change out the shocks and add air springs. I did all of the above on my 2016 F150 and it is nearly as stable as a 2019 F250 that I had for a while plus I can let the air out of the air springs and have a softer ride when empty.
I understand your frustration with finding a max payload package on the lot. I have never seen one. However, when I was asking about the max payload package someone on this site did offer to let me drive theirs to feel the difference. It was a bit of a long drive so I didn't take them up on it but you might be luckier. You might want give a better idea of where you live in Texas if you want to try for that. As much as Ford like to advertise having the largest payload of any half ton truck they should do a lot more to help people buy it.
